Problem: $ D = \left[\begin{array}{r}2 \\ -2 \\ 1\end{array}\right]$ $ E = \left[\begin{array}{rr}-2 & 3\end{array}\right]$ What is $ D E$ ?
Solution: Because $ D$ has dimensions $(3\times1)$ and $ E$ has dimensions $(1\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ D E = \left[\begin{array}{r}{2} \\ {-2} \\ \color{gray}{1}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{3}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ E$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ E$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ E$ , and so on. Add the products together. $ \left[\begin{array}{rr}{2}\cdot{-2} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{-2} & ? \\ {-2}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ E$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{-2} & {2}\cdot\color{#DF0030}{3} \\ {-2}\cdot{-2} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{2}\cdot{-2} & {2}\cdot\color{#DF0030}{3} \\ {-2}\cdot{-2} & {-2}\cdot\color{#DF0030}{3} \\ \color{gray}{1}\cdot{-2} & \color{gray}{1}\cdot\color{#DF0030}{3}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-4 & 6 \\ 4 & -6 \\ -2 & 3\end{array}\right] $